Prove that there exists a sequence of unbounded positive integers $a_1\leq a_2\leq a_3\leq\cdots$, such that there exists a positive integer $M$ with the following property: for any integer $n\geq M$, if $n+1$ is not prime, then any prime divisor of $n!+1$ is greater than $n+a_n$.

Does there exist two infinite sets $A,B$ such that every number can be written uniquely as a sum of an element of $A$ and an element of $B$?

A circumcircle of triangle $ABC$ meets the sides $AD$ and $CD$ of a parallelogram $ABCD$ at points $K$ and $L$ respectively. Let $M$ be the midpoint of arc $KL$ not containing $B$. Prove that $DM \perp AC$. by E.Bakaev

Let it \(k\) be a fixed positive integer. Alberto and Beralto play the following game: Given an initial number \(N_0\) and starting with Alberto, they alternately do the following operation: change the number \(n\) for a number \(m\) such that \(m < n\) and \(m\) and \(n\) differ, in its base-2 representation, in exactly \(l\) consecutive digits for some \(l\) such that \(1 \leq l \leq k\). If someone can't play, he loses. We say a non-negative integer \(t\) is a [i]winner[/i] number when the gamer who receives the number \(t\) has a winning strategy, that is, he can choose the next numbers in order to guarrantee his own victory, regardless the options of the other player. Else, we call it [i]loser[/i]. Prove that, for every positive integer \(N\), the total of non-negative loser integers lesser than \(2^N\) is \(2^{N-\lfloor \frac{log(min\{N,k\})}{log 2} \rfloor}\)

For how many real values of $x$ is the equation $(x^2 - 7)^3 = 0$ true? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$

In the spherical shaped planet $X$ there are $2n$ gas stations. Every station is paired with one other station , and every two paired stations are diametrically opposite points on the planet. Each station has a given amount of gas. It is known that : if a car with empty (large enough) tank starting from any station it is always to reach the paired station with the initial station (it can get extra gas during the journey). Find all naturals $n$ such that for any placement of $2n$ stations for wich holds the above condotions, holds: there always a gas station wich the car can start with empty tank and go to all other stations on the planet.(Consider that the car consumes a constant amount of gas per unit length.)

Let $a,b,c$ be nonnegative real numbers such that $(a+b)(b+c)(c+a) \neq0$. Find the minimum of $(a+b+c)^{2016}(\frac{1}{a^{2016}+b^{2016}}+\frac{1}{b^{2016}+c^{2016}}+\frac{1}{c^{2016}+a^{2016}})$.

Let $ABCD$ be a parallelogram, such that the point $M$ is the midpoint of the side $CD$ and lies on the bisector of the angle $\angle BAD$. Prove that $\angle AMB = 90^o$.

Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc \equal{} 1$. Prove that \[ \frac {1}{a^{3}\left(b \plus{} c\right)} \plus{} \frac {1}{b^{3}\left(c \plus{} a\right)} \plus{} \frac {1}{c^{3}\left(a \plus{} b\right)}\geq \frac {3}{2}. \]

The positive integers $a, b, c, d$ satisfy (i) $a + b + c + d = 2023$ (ii) $2023 \text{ } | \text{ } ab - cd$ (iii) $2023 \text{ } | \text{ } a^2 + b^2 + c^2 + d^2.$ Assuming that each of the numbers $a, b, c, d$ is divisible by $7$, prove that each of the numbers $a, b, c, d$ is divisible by $17$.

Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$. [i]South Africa [/i]

[b]Problem 2[/b] Consider the infinite chessboard whose rows and columns are indexed by positive integers. Is it possible to put a single positive rational number into each cell of the chessboard so that each positive rational number appears exactly once and the sum of every row and of every column is finite?

A line $d$ is called [i]faithful[/i] to triangle $ABC$ if $d$ be in plane of triangle $ABC$ and the reflections of $d$ over the sides of $ABC$ be concurrent. Prove that for any two triangles with acute angles lying in the same plane, either there exists exactly one [i]faithful[/i] line to both of them, or there exist infinitely [i]faithful[/i] lines to them.

Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

Find all integer-coefficient polynomials $f(x)$ satisfying the following conditions for every integer $n \geqslant 2$: [list] [*] $f(n) > 0$. [*] $f(n)$ divides $n^{f(n)} - 1$. [/list]

[i]20 problems for 20 minutes. [/i] [b]p1.[/b] Evaluate $\frac{\sqrt2 \cdot \sqrt6}{\sqrt3}.$ [b]p2.[/b] If $6\%$ of a number is $1218$, what is $18\%$ of that number? [b]p3.[/b] What is the median of $\{42, 9, 8, 4, 5, 1,13666, 3\}$? [b]p4.[/b] Define the operation $\heartsuit$ so that $i\heartsuit u = 5i - 2u$. What is $3\heartsuit 4$? p5. How many $0.2$-inch by $1$-inch by $1$-inch gold bars can fit in a $15$-inch by $12$-inch by $9$-inch box? [b]p6.[/b] A tetrahedron is a triangular pyramid. What is the sum of the number of edges, faces, and vertices of a tetrahedron? [b]p7.[/b] Ron has three blue socks, four white socks, five green socks, and two black socks in a drawer. Ron takes socks out of his drawer blindly and at random. What is the least number of socks that Ron needs to take out to guarantee he will be able to make a pair of matching socks? [b]p8.[/b] One segment with length $6$ and some segments with lengths $10$, $8$, and $2$ form the three letters in the diagram shown below. Compute the sum of the perimeters of the three figures. [img]https://cdn.artofproblemsolving.com/attachments/1/0/9f7d6d42b1d68cd6554d7d5f8dd9f3181054fa.png[/img] [b]p9.[/b] How many integer solutions are there to the inequality $|x - 6| \le 4$? [b]p10.[/b] In a land for bad children, the flavors of ice cream are grass, dirt, earwax, hair, and dust-bunny. The cones are made out of granite, marble, or pumice, and can be topped by hot lava, chalk, or ink. How many ice cream cones can the evil confectioners in this ice-cream land make? (Every ice cream cone consists of one scoop of ice cream, one cone, and one topping.) [b]p11.[/b] Compute the sum of the prime divisors of $245 + 452 + 524$. [b]p12.[/b] In quadrilateral $SEAT$, $SE = 2$, $EA = 3$, $AT = 4$, $\angle EAT = \angle SET = 90^o$. What is the area of the quadrilateral? [b]p13.[/b] What is the angle, in degrees, formed by the hour and minute hands on a clock at $10:30$ AM? [b]p14.[/b] Three numbers are randomly chosen without replacement from the set $\{101, 102, 103,..., 200\}$. What is the probability that these three numbers are the side lengths of a triangle? [b]p15.[/b] John takes a $30$-mile bike ride over hilly terrain, where the road always either goes uphill or downhill, and is never flat. If he bikes a total of $20$ miles uphill, and he bikes at $6$ mph when he goes uphill, and $24$ mph when he goes downhill, what is his average speed, in mph, for the ride? [b]p16.[/b] How many distinct six-letter words (not necessarily in any language known to man) can be formed by rearranging the letters in $EXETER$? (You should include the word EXETER in your count.) [b]p17.[/b] A pie has been cut into eight slices of different sizes. Snow White steals a slice. Then, the seven dwarfs (Sneezy, Sleepy, Dopey, Doc, Happy, Bashful, Grumpy) take slices one by one according to the alphabetical order of their names, but each dwarf can only take a slice next to one that has already been taken. In how many ways can this pie be eaten by these eight persons? [b]p18.[/b] Assume that $n$ is a positive integer such that the remainder of $n$ is $1$ when divided by $3$, is $2$ when divided by $4$, is $3$ when divided by $5$, $...$ , and is $8$ when divided by $10$. What is the smallest possible value of $n$? [b]p19.[/b] Find the sum of all positive four-digit numbers that are perfect squares and that have remainder $1$ when divided by $100$. [b]p20.[/b] A coin of radius $1$ cm is tossed onto a plane surface that has been tiled by equilateral triangles with side length $20\sqrt3$ cm. What is the probability that the coin lands within one of the triangles? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Write $\displaystyle{\frac{1}{\sqrt[5]{2} - 1} = a + b\sqrt[5]{2} + c\sqrt[5]{4} + d\sqrt[5]{8} + e\sqrt[5]{16}}$, with $a$, $b$, $c$, $d$, and $e$ integers. Find $a^2 + b^2 + c^2 + d^2 + e^2$.

The sphere inscribed in the tetrahedron $ABCD$ touches its faces at points $A',B',C',D'$. The segments $AA'$ and $BB'$ intersect, and the point of their intersection lies on the inscribed sphere. Prove that the segments $CC'$ and $DD'$ also intersect on the inscribed sphere.

Three circles are pairwise tangent, with none of them lying inside another. The centres of the circles are the corners of a triangle with circumference $1$. What is the smallest possible value for the sum of the areas of the circles?

Prove that every natural number is the difference of two natural numbers that have the same number of prime factors. (Each prime divisor is counted once, for example, the number $12$ has two prime factors: $2$ and $3$.)

There are $4n$ pebbles of weights $1, 2, 3, \dots, 4n.$ Each pebble is coloured in one of $n$ colours and there are four pebbles of each colour. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied: [list] [*]The total weights of both piles are the same. [*] Each pile contains two pebbles of each colour. [/list] [i]Proposed by Milan Haiman, Hungary and Carl Schildkraut, USA[/i]

Given is a non-isosceles triangle $ABC$ with $\angle ABC=60^{\circ}$, and in its interior, a point $T$ is selected such that $\angle ATC= \angle BTC=\angle BTA=120^{\circ}$. Let $M$ the intersection point of the medians in $ABC$. Let $TM$ intersect $(ATC)$ at $K$. Find $TM/MK$.

At the vertices of a regular hexagon are written six nonnegative integers whose sum is $2003^{2003}$. Bert is allowed to make moves of the following form: he may pick a vertex and replace the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after which the number 0 appears at all six vertices.

For a positive integer $ n,$ $ a_{n}\equal{}n\sqrt{5}\minus{} \lfloor n\sqrt{5}\rfloor$. Compute the maximum value and the minimum value of $ a_{1},a_{2},\ldots ,a_{2009}.$

What is the positive difference between the sum of the first $5$ positive even integers and the first $5$ positive odd integers? $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$